Find the particular solution of the differential equation: \[ \frac{dy}{dx} + y \cot x = 4x \csc x \text{(} x \neq 0 \text{)}. \] Given that \( y = 0 \) \(\text{ when}\) \( x = \frac{\pi}{2} \).
Find the vector equation of a plane which passes through the point of intersection of the planes \[ \vec{r} \cdot (\hat{i} + \hat{j} + \hat{k}) = 6 \, \text{and} \, \vec{r} \cdot (2\hat{i} + 3\hat{j} + 4\hat{k}) = -5 \] and the point $(1,1,1)$.
A relation \( R = \{(x, y) : \text{Number of pages in} \, x \text{ and } y \text{ are equal} \} \) is defined on the set \( A \) of all books in a college library. Prove that \( R \) is an equivalence relation.
Find the equation of the plane which passes through the intersecting point of the planes \[ \vec{r} \cdot (\hat{i} + \hat{j} + \hat{k}) = 6 \, \text{and} \, \vec{r} \cdot (2\hat{i} + 3\hat{j} + 4\hat{k}) = -5, \] and the point \( (1, 1, 1) \).
Show that \( f(x) = |x| \, \textbf{is continuous at} \, x = 0.\)
If \[ P(A) = \frac{3}{13}, P(B) = \frac{5}{13}, \text{and} P(A \cap B) = \frac{2}{13}, \] \(\text{then find the value of }\) \( P(B|A) \).
If \[ P(A) = 0.4 \, \text{and} \, P(B) = 0.5, \, \text{also, A and B are independent events, then find} \] (i) \( P(A \cup B) \) and (ii) \( P(A \cap B) \).
Find the minimum value of \[ Z = 50x + 70y \] \(\text{under the following constraints by graphical method:}\) \[ 2x + y \geq 8, \] \[ x + 2y \geq 10, x \geq 0, y \geq 0. \]
If \[ A = \begin{bmatrix} 1 & 3 & 3 \\ 1 & 4 & 3 \\ 1 & 3 & 4 \end{bmatrix}, \] \(\text{then prove that}\) \[ A \cdot \text{adj}(A) = |A| \cdot I. \text{ Also, find } A^{-1}. \]